Plotting the Quadratic Function Using Parameters a, b and c

Plotting the Quadratic Function

Plotting the graph of the quadratic function and examining the roles of the parameters $a, b, c$ in the function of the form $y = ax^2 + bx + c$

The quadratic function has three relevant characteristics:

$a$ – the coefficient of $X^2$ .
$b$ – the coefficient of $X$ .
$c$ – the constant term.

Steps to graph a quadratic function –

Let's examine the parameter $a$ and ask: Is the function upward or downward facing?
Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
Let's find the points of intersection with the $X$ axis by substituting ( $Y=0$ ).
Let's draw a coordinate system and first mark the vertex of the parabola.
Then, let's examine if the function is smiling or crying and mark the points of intersection with the $X$ axis that we found. Draw accordingly.

Detailed explanation

Start practice

Test yourself on plotting functions with parameters!

What is the value of the coefficient $$ b $$ in the equation below?

$$ 3x^2+8x-5 $$

Practice more now

Graphing the quadratic function ausing parameters a, b and c

Let's present the quadratic function and understand the meaning of each parameter in it.
$y=ax^2+bx+c$

$a$ – the coefficient of $X^2$
- -determines if the parabola will be a maximum or minimum parabola (sad or smiling) - determines the steepness – its width.

$a$ must be different from 0.
If $a$ is positive – the parabola is a minimum parabola – smiling
If $a$ is negative – the parabola is a maximum parabola – sad
The larger $a$ is – the narrower the function will be and vice versa.

$b$ - the coefficient of $X$
can be any number
indicates the position of the parabola along with $C$ .
determines the slope of the function at the intersection point with the $Y$ axis
(not relevant to the material)

$c$ – the constant term
is not dependent on $X$
can be any number
indicates the position of the parabola along with $X$
determines the y-intercept
responsible for the vertical shift of the function

Wonderful. Now, let's move on to plotting the quadratic function.

Steps to graph a quadratic function –

Let's examine the parameter $a$ and ask: is the function upward or downward facing?
Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
Let's find the points of intersection with the $X$ axis by substituting ( $Y=0$ ).
Let's draw a coordinate system and first mark the vertex of the parabola.
Then, let's examine if the function is smiling or sad and mark the points of intersection with the $X$ axis that we found. Draw accordingly.

Let's see an example -

In the following function:
$Y=-4X^2+4x+3$

$a=-4$ , negative. Therefore, the function is downward facing.
Let's find the vertex of the parabola:
$X=\frac{-4}{-4*2}=\frac{-4}{-8}=\frac{1}{2}$
$y=-4*\frac{1^2}{2}+4*\frac{1}{2}+3$
$y=4$
Vertex of the parabola $(\frac{1}{2}, 4)$
Let's find the intersection points with the $X$ axis
Substitute
$Y=0$
and we get:
$-4X^2+4x+3=0$
$X=-\frac{1}{2},1\frac{1}{2}$
Draw a coordinate system, mark relevant points, and draw logically:

Join Over 30,000 Students Excelling in Math!

Endless Practice, Expert Guidance - Elevate Your Math Skills Today

Test your knowledge

Question 1

What is the value of the coefficient $$ c $$ in the equation below?

$$ 4x^2+9x-2 $$

Question 2

Identify the coefficients based on the following equation

$$ y=-2x^2+3x+10 $$

Question 3

Identify the coefficients based on the following equation

$$ y=x^2 $$

Examples with solutions for Plotting Functions with Parameters

Exercise #1

$y=x^2+10x$

Video Solution

Step-by-Step Solution

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

$y = ax²+bx+c$

Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.

Firstly, it seems that in this formula we do not have the C,

Therefore, we understand it is equal to 0.

$c = 0$

a is the coefficient of X², here it does not have a coefficient, therefore

$a = 1$

$b= 10$

is the number that comes before the X that is not squared.

Answer

$a=1,b=10,c=0$

Exercise #2

$y=2x^2-5x+6$

Video Solution

Step-by-Step Solution

In fact, a quadratic equation is composed as follows:

y = ax²-bx-c

That is,

a is the coefficient of x², in this case 2.
b is the coefficient of x, in this case 5.
And c is the number without a variable at the end, in this case 6.

Answer

$a=2,b=-5,c=6$

Exercise #3

Identify the coefficients based on the following equation

$y=-2x^2+3x+10$

Video Solution

Step-by-Step Solution

Let's determine the coefficients for the quadratic function given by $y = -2x^2 + 3x + 10$ .

Step 1: Identify $a$ .
The coefficient of $x^2$ is $-2$ . Thus, $a = -2$ .
Step 2: Identify $b$ .
The coefficient of $x$ is $3$ . Thus, $b = 3$ .
Step 3: Identify $c$ .
The constant term is $10$ . Thus, $c = 10$ .

Comparing these coefficients to the provided choices, the correct answer is:

$a = -2, b = 3, c = 10$ .

Therefore, the correct choice is Choice 4.

Answer

$a=-2,b=3,c=10$

Exercise #4

Determine the value of the coefficient $a$ in the following equation:

$-x^2+7x-9$

Video Solution

Step-by-Step Solution

The quadratic equation in the problem is already arranged (meaning all terms are on one side and 0 on the other side), so let's proceed to answer the question asked:

The question asked in the problem - What is the value of the coefficient $a$ in the equation?

Let's recall the definitions of coefficients in solving quadratic equations and the roots formula:

The rule states that the roots of an equation of the form:

$ax^2+bx+c=0$ are:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

That is the coefficient $a$ is the coefficient of the quadratic term (meaning the term with the second power)- $x^2$ Let's examine the equation in the problem:

$-x^2+7x-9 =0$

Remember that the minus sign before the quadratic term means multiplication by: $-1$ , therefore- we can write the equation as:

$-1\cdot x^2+7x-9 =0$

The number that multiplies the $x^2$ , is $-1$ hence we identify that the coefficient of the quadratic term is the number $-1$ ,

Therefore the correct answer is A.

Answer

-1

Exercise #5

What is the value of the coefficient $c$ in the equation below?

$4x^2+9x-2$

Video Solution

Step-by-Step Solution

The quadratic equation is given as $4x^2 + 9x - 2$ . This equation is in the standard form of a quadratic equation, which is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are coefficients.

The term $4x^2$ indicates that the coefficient $a = 4$ .
The term $9x$ indicates that the coefficient $b = 9$ .
The constant term $-2$ indicates that the coefficient $c = -2$ .

From this analysis, we can see that the coefficient $c$ is $-2$ .

Therefore, the value of the coefficient $c$ in the equation is $-2$ .

Answer

-2

Start practice

Plotting the Quadratic Function Using Parameters a, b and c